Bipartite characterization of rational POP generating functions

Let a POP class be a permutation class defined by avoidance of partially ordered patterns. Its generating function is the ordinary generating function of its counting sequence, and the class is bipartite when its defining partially ordered pattern has a bipartite underlying poset.

Bipartite characterization conjecture. A POP class has a rational generating function if and only if it is bipartite.

The conjecture strengthens the computational observation that the size-$5$ POP classes found with rational generating functions were bipartite and had regular insertion encodings. The supplied context notes that the available regular-insertion-encoding results do not prove that non-regular classes cannot have rational generating functions.